From p-adic to real Grassmannians via the quantum

Details From p-adic to real Grassmannians via the quantum From p-adic to real Grassmannians via the quantum

2004-05-07

arxiv.org/abs/math/0405138v4

Mathematics

Representation Theory

24 pages, final version, to appear in Advances in Mathematics

Scientific paper

Let F be a local field. The action of GL(n,F) on the Grassmann variety Gr(m,n,F) induces a continuous representation of the maximal compact subgroup of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible constituents of this representation are parameterized by the same underlying set both for Archimedean and non-Archimedean fields. This paper connects the Archimedean and non-Archimedean theories using the quantum Grassmannian. In particular, idempotents in the Hecke algebra associated to this representation are the image of the quantum zonal spherical functions after taking appropriate limits. Consequently, a correspondence is established between some irreducible representations with Archimedean and non-Archimedean origin.

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